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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution:</dfn> Seek a solution of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y=e^{rx}.
\end{equation*}
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<p class="continuation">Substituting it into the ODE,</p>
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\begin{equation*}
2 r^2 e^{rx}-3 r e^{rx}+e^{rx}=0 ~\rightarrow~2 r^2-3r+1=0~\rightarrow~(2r-1)(r-1)=0~\rightarrow~r=\frac{1}{2}~ \textrm{or} ~1.
\end{equation*}
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<p class="continuation">Thus, there are two solutions:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y_1=e^{\frac{1}{2}x},\quad y_2=e^x.
\end{equation*}
</div>
<p class="continuation">The general solution is</p>
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\begin{equation*}
y=C_1 e^{\frac{1}{2}x}+C_2 e^x,
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(C_1\)</span> and <span class="process-math">\(C_2\)</span> are arbitrary constants.</p>
<span class="incontext"><a href="sec3_1.html#p-69" class="internal">in-context</a></span>
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